Its not just the firmware...
The driver package can (and sometimes does) also prevent you from using non-HP branch ink.
I have an HP MFP, with non-standard ink.
Worked fine in Windows 7 with their ancient HP driver package.
Upgraded to Windows 10, and pulled the HP app from the windows store.
The App STOPS you from printing, cause of non-HP ink!
Tried to uninstall the app - HA! you can't!
Tried to delete the app where it is installed on HDD - HA! not even administrator can reach that director (not without some reg hack to allow ownership...)
Managed to get rid of the app after some research, and installed the ancient Windows 7 driver package... works fine.